Question

A stationary particle explodes into two particles of masses $m_1$ and $m_2$ which move in opposite directions with velocities $v_1$ and $v_2$. The ratio of their kinetic energies $E_1 /E_2$ is

• A. $1$
• B. $m_{1}v_{2}/m_{2}v_{1}$
• C. $m_{2}/m_{1}$
• D. $m_{1}/m_{2}$

Answer 1

Answer: (C)

$m_{2}/m_{1}$

Answer 2

For a exploding body, linear momentum is conserved.
From conservation of linear momentum
$P_{\text {initial }}=P_{\text {final }}$
$0=m_{1} v_{1}-m_{2} v_{2}$
or $m_{1} v_{1}=m_{2} v_{2}$
or $\frac{v_{1}}{v_{2}}=\frac{m_{2}}{m_{1}}\,\,\,$...(i)
Thus, ratio of kinetic energies
$\frac{E_{1}}{E_{2}}=\frac{\frac{1}{2} m_{1} v_{1}^{2}}{\frac{1}{2} m_{2} v_{2}^{2}}=\frac{m_{1}}{m_{2}} \times\left(\frac{m_{2}}{m_{1}}\right)^{2}$
$=\frac{m_{2}}{m_{1}}$